Integral homology 3-spheres and the Johnson filtration

Pitsch, Wolfgang

doi:10.1090/S0002-9947-08-04208-6

Integral homology -spheres and the Johnson filtration

Author: Wolfgang Pitsch
Journal: Trans. Amer. Math. Soc. 360 (2008), 2825-2847
MSC (2000): Primary 57M99; Secondary 20F38, 20F12
DOI: https://doi.org/10.1090/S0002-9947-08-04208-6
Published electronically: January 4, 2008
MathSciNet review: 2379777
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The mapping class group of an oriented surface $\Sigma_{g,1}$ of genus with one boundary component has a natural decreasing filtration $\mathcal{M}_{g,1} \supset \mathcal{M}_{g,1}(1) \supset \mathcal{M}_{g,1}(2) \supset \mathcal{M}_{g,1}(3) \supset \cdots$ , where $\mathcal{M}_{g,1}(k)$ is the kernel of the action of $\mathcal{M}_{g,1}$ on the $k^{th}$ nilpotent quotient of $\pi_1(\Sigma_{g,1})$ . Using a tree Lie algebra approximating the graded Lie algebra $\displaystyle \bigoplus_{k} \mathcal{M}_{g,1}(k)/\mathcal{M}_{g,1}(k+1)$ we prove that any integral homology sphere of dimension has for some a Heegaard decomposition of the form $M = \mathcal{H}_g \coprod_{\iota_g \phi} - \mathcal{H}_g$ , where $\phi \in \mathcal{M}_{g,1}(3)$ and $\iota_g$ is such that $\mathcal{H}_g \coprod_{\iota_g} - \mathcal{H}_g= S^3$ . This proves a conjecture due to S. Morita and shows that the ``core'' of the Casson invariant is indeed the Casson invariant.

1. Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. MR 1318886, https://doi.org/10.1016/0040-9383(95)93237-2
2. Robert Craggs, A new proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings, Proc. Amer. Math. Soc. 57 (1976), no. 1, 143–147. MR 0410749, https://doi.org/10.1090/S0002-9939-1976-0410749-9
3. Stavros Garoufalidis and Jerome Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005, pp. 173–203. MR 2131016, https://doi.org/10.1090/pspum/073/2131016
4. H. B. Griffiths, Automorphisms of a 3-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg 26 (1963/1964), 191–210. MR 0159313, https://doi.org/10.1007/BF02992786
5. Nathan Habegger and Gregor Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), no. 6, 1253–1289. MR 1783857, https://doi.org/10.1016/S0040-9383(99)00041-5
6. Nathan Habegger and Wolfgang Pitsch, Tree level Lie algebra structures of perturbative invariants, J. Knot Theory Ramifications 12 (2003), no. 3, 333–345. MR 1983089, https://doi.org/10.1142/S0218216503002494
7. Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
8. Dennis Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985), no. 2, 113–126. MR 793178, https://doi.org/10.1016/0040-9383(85)90049-7
9. Jerome Levine, Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group” [Algebr. Geom. Topol. 1 (2001), 243–270; MR1823501 (2002m:57020)], Algebr. Geom. Topol. 2 (2002), 1197–1204. MR 1943338, https://doi.org/10.2140/agt.2002.2.1197
Jerome Levine, Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001), 243–270. MR 1823501, https://doi.org/10.2140/agt.2001.1.243
10. Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, 2nd ed., Dover Publications, Inc., Mineola, NY, 2004. Presentations of groups in terms of generators and relations. MR 2109550
11. Shigeyuki Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. I, Topology 28 (1989), no. 3, 305–323. MR 1014464, https://doi.org/10.1016/0040-9383(89)90011-6
12. Shigeyuki Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), no. 3, 699–726. MR 1224104, https://doi.org/10.1215/S0012-7094-93-07017-2
13. J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen i, Acta Math 50 (1927), 189-358.
14. -, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen II, Acta Math 53 (1929), 1-76.
15. -, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen III, Acta Math 58 (1931), 87-167.
16. Wolfgang Pitsch, Une construction intrinsèque du cœur de l’invariant de Casson, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1741–1761 (French, with English and French summaries). MR 1871288
17. Wolfgang Pitsch, Extensions verselles et automorphismes des groupes nilpotents libres, J. Algebra 249 (2002), no. 2, 512–527 (French, with French summary). MR 1901170, https://doi.org/10.1006/jabr.2001.9077
18. K. Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg (1933), 189-194.
19. James Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), no. 1, 88–111. MR 1501673, https://doi.org/10.1090/S0002-9947-1933-1501673-5
20. Shin’ichi Suzuki, On homeomorphisms of a 3-dimensional handlebody, Canad. J. Math. 29 (1977), no. 1, 111–124. MR 0433433, https://doi.org/10.4153/CJM-1977-011-1